## Defining parameters

 Level: $$N$$ = $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$36$$ Sturm bound: $$691200$$ Trace bound: $$28$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(2550))$$.

Total New Old
Modular forms 176384 39542 136842
Cusp forms 169217 39542 129675
Eisenstein series 7167 0 7167

## Trace form

 $$39542q - 6q^{2} - 14q^{3} - 6q^{4} - 20q^{5} - 14q^{6} - 48q^{7} - 6q^{8} - 6q^{9} + O(q^{10})$$ $$39542q - 6q^{2} - 14q^{3} - 6q^{4} - 20q^{5} - 14q^{6} - 48q^{7} - 6q^{8} - 6q^{9} - 4q^{10} - 40q^{11} + 10q^{12} - 52q^{13} - 16q^{14} + 8q^{15} - 14q^{16} + 2q^{17} + 14q^{18} + 72q^{19} + 16q^{20} + 16q^{21} + 88q^{22} + 80q^{23} + 18q^{24} + 172q^{25} - 12q^{26} - 14q^{27} + 32q^{28} + 36q^{29} + 40q^{30} - 32q^{31} + 14q^{32} + 8q^{33} + 28q^{34} + 16q^{35} - 6q^{36} - 16q^{37} - 24q^{38} + 108q^{39} + 12q^{40} - 52q^{41} + 64q^{42} + 104q^{43} - 40q^{44} + 140q^{45} + 48q^{46} + 64q^{47} + 18q^{48} + 126q^{49} - 36q^{50} + 94q^{51} - 20q^{52} + 40q^{53} + 42q^{54} + 48q^{55} - 16q^{56} + 112q^{57} - 116q^{58} - 72q^{59} - 64q^{60} - 196q^{61} - 96q^{62} - 168q^{63} - 6q^{64} - 148q^{65} - 88q^{66} - 248q^{67} + 50q^{68} - 168q^{69} + 160q^{70} + 368q^{71} + 18q^{72} + 892q^{73} + 436q^{74} - 72q^{75} + 8q^{76} + 672q^{77} + 212q^{78} + 736q^{79} + 108q^{80} - 46q^{81} + 828q^{82} + 600q^{83} + 88q^{84} + 734q^{85} + 408q^{86} + 596q^{87} + 408q^{88} + 664q^{89} - 76q^{90} + 1344q^{91} + 176q^{92} + 552q^{93} + 736q^{94} + 480q^{95} + 18q^{96} + 836q^{97} + 354q^{98} + 224q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(2550))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
2550.2.a $$\chi_{2550}(1, \cdot)$$ 2550.2.a.a 1 1
2550.2.a.b 1
2550.2.a.c 1
2550.2.a.d 1
2550.2.a.e 1
2550.2.a.f 1
2550.2.a.g 1
2550.2.a.h 1
2550.2.a.i 1
2550.2.a.j 1
2550.2.a.k 1
2550.2.a.l 1
2550.2.a.m 1
2550.2.a.n 1
2550.2.a.o 1
2550.2.a.p 1
2550.2.a.q 1
2550.2.a.r 1
2550.2.a.s 1
2550.2.a.t 1
2550.2.a.u 1
2550.2.a.v 1
2550.2.a.w 1
2550.2.a.x 1
2550.2.a.y 1
2550.2.a.z 1
2550.2.a.ba 1
2550.2.a.bb 1
2550.2.a.bc 1
2550.2.a.bd 1
2550.2.a.be 1
2550.2.a.bf 1
2550.2.a.bg 2
2550.2.a.bh 2
2550.2.a.bi 2
2550.2.a.bj 2
2550.2.a.bk 2
2550.2.a.bl 2
2550.2.a.bm 2
2550.2.a.bn 3
2550.2.a.bo 3
2550.2.c $$\chi_{2550}(1801, \cdot)$$ 2550.2.c.a 2 1
2550.2.c.b 2
2550.2.c.c 2
2550.2.c.d 2
2550.2.c.e 2
2550.2.c.f 2
2550.2.c.g 2
2550.2.c.h 2
2550.2.c.i 2
2550.2.c.j 2
2550.2.c.k 2
2550.2.c.l 4
2550.2.c.m 4
2550.2.c.n 4
2550.2.c.o 4
2550.2.c.p 4
2550.2.c.q 4
2550.2.c.r 6
2550.2.c.s 6
2550.2.d $$\chi_{2550}(2449, \cdot)$$ 2550.2.d.a 2 1
2550.2.d.b 2
2550.2.d.c 2
2550.2.d.d 2
2550.2.d.e 2
2550.2.d.f 2
2550.2.d.g 2
2550.2.d.h 2
2550.2.d.i 2
2550.2.d.j 2
2550.2.d.k 2
2550.2.d.l 2
2550.2.d.m 2
2550.2.d.n 2
2550.2.d.o 2
2550.2.d.p 2
2550.2.d.q 2
2550.2.d.r 2
2550.2.d.s 2
2550.2.d.t 2
2550.2.d.u 4
2550.2.d.v 4
2550.2.f $$\chi_{2550}(1699, \cdot)$$ 2550.2.f.a 2 1
2550.2.f.b 2
2550.2.f.c 2
2550.2.f.d 2
2550.2.f.e 2
2550.2.f.f 2
2550.2.f.g 2
2550.2.f.h 2
2550.2.f.i 2
2550.2.f.j 2
2550.2.f.k 2
2550.2.f.l 2
2550.2.f.m 2
2550.2.f.n 2
2550.2.f.o 4
2550.2.f.p 4
2550.2.f.q 4
2550.2.f.r 4
2550.2.f.s 4
2550.2.f.t 4
2550.2.i $$\chi_{2550}(1007, \cdot)$$ n/a 216 2
2550.2.l $$\chi_{2550}(443, \cdot)$$ n/a 192 2
2550.2.m $$\chi_{2550}(1849, \cdot)$$ n/a 104 2
2550.2.p $$\chi_{2550}(1951, \cdot)$$ n/a 116 2
2550.2.q $$\chi_{2550}(407, \cdot)$$ n/a 216 2
2550.2.t $$\chi_{2550}(293, \cdot)$$ n/a 216 2
2550.2.u $$\chi_{2550}(511, \cdot)$$ n/a 320 4
2550.2.v $$\chi_{2550}(151, \cdot)$$ n/a 224 4
2550.2.x $$\chi_{2550}(257, \cdot)$$ n/a 432 4
2550.2.ba $$\chi_{2550}(593, \cdot)$$ n/a 432 4
2550.2.bc $$\chi_{2550}(49, \cdot)$$ n/a 224 4
2550.2.be $$\chi_{2550}(409, \cdot)$$ n/a 320 4
2550.2.bf $$\chi_{2550}(271, \cdot)$$ n/a 352 4
2550.2.bj $$\chi_{2550}(169, \cdot)$$ n/a 368 4
2550.2.bl $$\chi_{2550}(7, \cdot)$$ n/a 432 8
2550.2.bn $$\chi_{2550}(401, \cdot)$$ n/a 912 8
2550.2.bp $$\chi_{2550}(299, \cdot)$$ n/a 864 8
2550.2.bq $$\chi_{2550}(193, \cdot)$$ n/a 432 8
2550.2.bt $$\chi_{2550}(353, \cdot)$$ n/a 1440 8
2550.2.bv $$\chi_{2550}(203, \cdot)$$ n/a 1440 8
2550.2.bw $$\chi_{2550}(361, \cdot)$$ n/a 704 8
2550.2.bz $$\chi_{2550}(259, \cdot)$$ n/a 736 8
2550.2.ca $$\chi_{2550}(137, \cdot)$$ n/a 1280 8
2550.2.cc $$\chi_{2550}(47, \cdot)$$ n/a 1440 8
2550.2.ce $$\chi_{2550}(19, \cdot)$$ n/a 1408 16
2550.2.ch $$\chi_{2550}(263, \cdot)$$ n/a 2880 16
2550.2.ci $$\chi_{2550}(53, \cdot)$$ n/a 2880 16
2550.2.cl $$\chi_{2550}(121, \cdot)$$ n/a 1472 16
2550.2.cn $$\chi_{2550}(37, \cdot)$$ n/a 2880 32
2550.2.cp $$\chi_{2550}(29, \cdot)$$ n/a 5760 32
2550.2.cr $$\chi_{2550}(11, \cdot)$$ n/a 5760 32
2550.2.cs $$\chi_{2550}(73, \cdot)$$ n/a 2880 32

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(2550))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(2550)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(34))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(51))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(85))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(102))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(170))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(255))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(425))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(510))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(850))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1275))$$$$^{\oplus 2}$$